2 research outputs found
Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations
Two numerical methods with graded temporal grids are analyzed for fractional
evolution equations. One is a low-order discontinuous Galerkin (DG)
discretization in the case of fractional order , and the other one
is a low-order Petrov Galerkin (PG) discretization in the case of fractional
order . By a new duality technique, pointwise-in-time error
estimates of first-order and -order temporal accuracies are
respectively derived for DG and PG, under reasonable regularity assumptions on
the initial value. Numerical experiments are performed to verify the
theoretical results
A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations
We introduce a symmetric fractional-order reduction (SFOR) method to
construct numerical algorithms on general nonuniform temporal meshes for
semilinear fractional diffusion-wave equations. By using the novel order
reduction method, the governing problem is transformed to an equivalent coupled
system, where the explicit orders of time-fractional derivatives involved are
all . The linearized L1 scheme and Alikhanov scheme
are then proposed on general time meshes. Under some reasonable regularity
assumptions and weak restrictions on meshes, the optimal convergence is derived
for the two kinds of difference schemes by energy method. An adaptive
time stepping strategy which based on the (fast linearized) L1 and Alikhanov
algorithms is designed for the semilinear diffusion-wave equations. Numerical
examples are provided to confirm the accuracy and efficiency of proposed
algorithms